Preference Kernel

Description: in the learning to rank scenario, the preference kernel (Shen and Joshi, 2003) compares two pairs of ordered objects and :

where BK is a generic kernel operating on the elements of the pairs. The underlying idea is to evaluate whether the first pair aligns better to the second pair in its regular order rather than to its inverted order.

Parameters:

baseKernel: the base kernel BK applied over the elements of the pairs.

Uncrossed Pairwise Sum Kernel

Description: it compares two pairs of ordered objects and , summing the contributions of the single element similarities:

where BK is a generic kernel operating on the elements of the pairs. It has been used in learning scenarios where the elements within a pair have different roles, such as text and hypothesis in Recognizing Textual Entailment (Filice et al., 2015), or question and answer in Question Answering (Filice et al., 2016).

Parameters:

baseKernel: the base kernel BK applied over the elements of the pairs.

intrapairSimProduct: whether adding or not to the kernel combination an extra term equivalent to the multiplication of the intra-pair similarities, i.e., .

Uncrossed Pairwise Product Kernel

Description: it compares two pairs of ordered objects and , multiplying the contributions of the single element similarities:

where BK is a generic kernel operating on the elements of the pairs. As for the UncrossedPairwiseSumKernel, it has been used in learning scenarios where the elements within a pair have different roles, such as text and hypothesis in Recognizing Textual Entailment (Filice et al., 2015), or question and answer in Question Answering (Filice et al., 2016). The product operation inherently applies a sort of logic and between the and .

Parameters:

baseKernel: the base kernel BK applied over the elements of the pairs.

intrapairSimProduct: whether adding or not to the kernel combination an extra term equivalent to the multiplication of the intra-pair similarities, i.e., .

Pairwise Sum Kernel

Description: it compares two pairs of objects and , summing the contributions of all pairwise similarities between the single elements:

where BK is a generic kernel operating on the elements of the pairs. It has been used in symmetric tasks, such as Paraphrase Identification, where the elements within a pair are interchangeable (Filice et al., 2015).

Parameters:

baseKernel: the base kernel BK applied over the elements of the pairs.

intrapairSimProduct: whether adding or not to the kernel combination an extra term equivalent to the multiplication of the intra-pair similarities, i.e., .

Pairwise Product Kernel

Description: it compares two pairs of objects and , summing the contributions of the two possible pairwise alignments:

where BK is a generic kernel operating on the elements of the pairs. It has been used in symmetric tasks, such as Paraphrase Identification, where the elements within a pair are interchangeable (Filice et al., 2015).

Parameters:

baseKernel: the base kernel BK applied over the elements of the pairs.

intrapairSimProduct: whether adding or not to the kernel combination an extra term equivalent to the multiplication of the intra-pair similarities, i.e., .

Best Pairwise Alignment Kernel

Description: it compares two pairs of objects and , evaluating the best pairwise alignment:

where BK is a generic kernel operating on the elements of the pairs, and softmax is a function put in place of the max operation, which would cause K not to be a valid kernel function (i.e., the resulting Gram matrix can violate the Mercer’s conditions). In particular, (c=100 is accurate enough). The BestPairwiseAlignmentKernel has been used in symmetric tasks, such as Paraphrase Identification (Filice et al., 2015), where the elements within a pair are interchangeable.

Parameters:

baseKernel: the base kernel BK applied over the elements of the pairs.

intrapairSimProduct: whether adding or not to the kernel combination an extra term equivalent to the multiplication of the intra-pair similarities, i.e., .

References

Simone Filice, Giovanni Da San Martino and Alessandro Moschitti. Relational Information for Learning from Structured Text Pairs. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics, ACL 2015.